Series: Reasoning Ability Notes for CUET PG

CUET PG – Reasoning Ability

Topic: Series

Series puzzles are a staple of reasoning tests.
They ask you to spot a pattern, fill a missing term, or predict the next term in a given sequence.

📗 Simple Explanation

A series is an ordered list of numbers (or objects) where each element follows a rule that can be expressed mathematically or logically.
Your job is to identify that rule and use it to:

Find a missing term (e.g., 2, 5, 10, ?, 34).
Predict the next term(s) (e.g., 3, 6, 12, 24, … → next term is 48).
Decide whether a proposed term fits the series (e.g., “Does 18 belong after 9 in the series 1, 3, 5, 9, 17, …?”).

📌 Key Points to Remember

Category	Definition	Typical Formula / Pattern	Quick Check
Arithmetic Series (AP)	Constant difference `d` between consecutive terms.	`aₙ = a₁ + (n‑1)d`	Compute two consecutive differences; they must be equal.
Geometric Series (GP)	Constant ratio `r` between consecutive terms.	`aₙ = a₁·rⁿ⁻¹`	Verify consecutive quotients are equal.
Fibonacci / Recursive	Each term is a sum (or other operation) of previous one or more terms.	`aₙ = aₙ₋₁ + aₙ₋₂`	Check if any term equals the sum of the two preceding terms.
Alternating / Mixed	Two or more interleaved sub‑series (e.g., odd/even positions).	Treat each sub‑series separately.	Split series into groups.
Square / Cube / Factorial	Terms are perfect squares, cubes, factorials, etc.	`n²`, `n³`, `n!`	Check if terms are perfect powers.
Other Patterns	- Incremental addition (1,3,6,10,15,…) = triangular numbers. - Multiplicative + additive (2,6,18,54,…) = `3×previous` + 0. - Alternating operations (+,–,×,÷).	Look for repeated operation sequences.	Identify the operation change and apply it.

Common Pitfalls

Assuming a single rule – some series combine two rules (e.g., AP on odd positions, GP on even).
Over‑fitting – a rule may fit the given terms but fail for the next term.
Ignoring the “most natural” pattern – choose the simplest mathematical operation that works.

🧮 Worked Example

Series: 3, 9, 27, ?, 243
Question: Find the missing term.

Step‑by‑Step

Look for a common ratio
- 9 ÷ 3 = 3
- 27 ÷ 9 = 3
- So the ratio r = 3.
Confirm it's a Geometric Series
- All consecutive quotients equal → GP.
Compute the missing term
- a₄ = a₃ × r = 27 × 3 = 81.
Verify with the next given term
- 243 ÷ 81 = 3 – works!

Answer: 81.

Another Example: Mixed Pattern

Series: 2, 4, 3, 6, 5, ?
Question: Find the missing term.

Analysis

Split into odd & even positions.
- Odd positions: 2, 3, 5 → Arithmetic with d = +1? No: 3-2 = 1, 5-3 = 2 → maybe Fibonacci? Actually it's 2, 3, 5 – looks like primes?
- Even positions: 4, 6, ? → Arithmetic with d = +2.
Even positions: 4 → 6 (+2). Next even term should be 6 + 2 = 8.
So missing term is 8.

Answer: 8.

📌 Quick Revision Summary

Pattern	How to Spot	Formula / Rule	Example
AP	Constant difference	`aₙ = a₁ + (n‑1)d`	`5, 9, 13, 17…` → d=4
GP	Constant ratio	`aₙ = a₁·rⁿ⁻¹`	`2, 6, 18, 54…` → r=3
Fibonacci	Each term = sum of two before	`aₙ = aₙ₋₁ + aₙ₋₂`	`1, 1, 2, 3, 5, 8…`
Alternating Series	Two interleaved subsequences	Treat each sub‑series separately	`1, 2, 4, 8, 16…` (AP on odd, GP on even)
Squares/Cubes	Perfect powers	`n²`, `n³`	`1, 4, 9, 16, 25…`
Triangular	Sum of first n integers	`Tₙ = n(n+1)/2`	`1, 3, 6, 10, 15…`
Other	Look for repeated operations	Identify operation pattern	`2, 5, 10, 17,…` (add 3, add 5, add 7…)

Quick Tips

Compute differences → AP test.
Compute quotients → GP test.
Check for prime, square, cube → simple look‑ups.
Split into groups if every second/third element behaves differently.
Always verify the rule with the next term (if given).